A question about uniform convergent series

Question

$X$ is a Banach space and $a_n:X\rightarrow \mathbb{R}$ is a sequence of (continuous) functions.
Suppose $\Sigma_{n=1}^\infty|a_n(x)|$ converges uniformly on $x\in B_X$, unit ball of $X$ (so we can find $N$ so that $\Sigma_{n=N}^\infty|a_n(x)|<\epsilon$ regardless of $x\in B_X$).
Is it true that for any sequence $(x_n)$ of $B_X$, the series $\Sigma_{n=1}^\infty|a_n(x_n)|$ converges?

Answer 1

No, and you don't need Banach spaces for a counterexample - $\mathbb{R}$ will do. Let $a_n$ be a piecewise linear function that is equal to $1/n$ at $2^{-n}$ and is zero outside the interval $(2^{-n-1}, 2^{-n+1})$. Then $\sum a_n$ converges uniformly, but $\sum a_n(2^{-n})$ diverges.

This is essentially asking whether the Weierstrass M-test is necessary for uniform convergence. It is sufficient but not necessary.